On the second eigenvalue of a Cayley graph of the symmetric group

In 2020, Siemons and Zalesski [On the second eigenvalue of some Cayley graphs of the symmetric group. {\it arXiv preprint arXiv:2012.12460}, 2020] determined the second eigenvalue of the Cayley graph $\Gamma_{n,k} = \operatorname{Cay}(\operatorname{Sym}(n), C(n,k))$ for $k = 0$ and $k=1$, where $C(n,k)$ is the conjugacy class of $(n-k)$-cycles. In this paper, it is proved that for any $n\geq 3$ and $k\in \mathbb{N}$ relatively small compared to $n$, the second eigenvalue of $\Gamma_{n,k}$ is the eigenvalue afforded by the irreducible character of $\operatorname{Sym}(n)$ that corresponds to the partition $[n-1,1]$. As a byproduct of our method, the result of Siemons and Zalesski when $k \in \{0,1\}$ is retrieved. Moreover, we prove that the second eigenvalue of $\Gamma_{n,n-5}$ is also equal to the eigenvalue afforded by the irreducible character of the partition $[n-1,1]$.